Question: Factor the following expression: $2$ $x^2$ $-15$ $x+$ $28$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(2)}{(28)} &=& 56 \\ {a} + {b} &=& & & {-15} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $56$ and add them together. The factors that add up to ${-15}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-8}$ $ \begin{eqnarray} {ab} &=& ({-7})({-8}) &=& 56 \\ {a} + {b} &=& {-7} + {-8} &=& -15 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {2}x^2 {-7}x {-8}x +{28} $ Group the terms so that there is a common factor in each group: $ ({2}x^2 {-7}x) + ({-8}x +{28}) $ Factor out the common factors: $ x(2x - 7) - 4(2x - 7) $ Notice how $(2x - 7)$ has become a common factor. Factor this out to find the answer. $(2x - 7)(x - 4)$